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Please tell me whether or not the 2 correspondences are upper hemicontinuous and PLEASE (using the definition) justify why.

1)F:[2,3]->R^2, F(r)={(x,y):abs(x)+abs(y)<=r}

2)F:R^n{0}->R^n, F(x)=B(x;||x||), the closed ball centred at x with radius ||x||.


abs=absolute value
is the complement
||x|| is distance

My definition of upper hemicontinuity is:

F:A->R^k, A contained in R^n, correspondence. I say F is upper hemicontinuous at p if: for all sequences {x^m} in A, {y^m} in R^k (where y^m belongs to F(x^m) for all m), and x^m->p belongs to A, y^m->q belongs to R^k, I have q belonging to F(p).

Solution Preview

1) Proof:
Consider any r in [2,3]. Let x_n->r as n->oo. Let y_n belongs to F(x_n) and y_n->q=(q1,q2). I want to show q belongs to F(r).
For any e>0, since x_n->r and y_n->q, then there exists some N>0, such that for all n>N, we have |x_n-r|<e and |y_n-q|<e. Then x_n<r+e. We know y_n is in F(x_n), Let ...

Solution Summary

This shows how to determine whether or not correspondences are upper hemicontinuous.